Solve (a-3)x+3b-4=(b-1)x+a+3

$$( a - 3 ) x + 3 b - 4 = ( b - 1 ) x + a + 3$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-bx-2x+3b-7}{x-1}\text{, }&x\neq 1\\a\in \mathrm{C}\text{, }&b=\frac{9}{2}\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-3 by x.

$$ax-3x+3b-4=\left(b-1\right)x+a+3$$

Use the distributive property to multiply b-1 by x.

$$ax-3x+3b-4=bx-x+a+3$$

Subtract a from both sides.

$$ax-3x+3b-4-a=bx-x+3$$

Add 3x to both sides.

$$ax+3b-4-a=bx-x+3+3x$$

Combine -x and 3x to get 2x.

$$ax+3b-4-a=bx+2x+3$$

Subtract 3b from both sides.

$$ax-4-a=bx+2x+3-3b$$

Add 4 to both sides.

$$ax-a=bx+2x+3-3b+4$$

Add 3 and 4 to get 7.

$$ax-a=bx+2x+7-3b$$

Combine all terms containing a.

$$\left(x-1\right)a=bx+2x+7-3b$$

The equation is in standard form.

$$\left(x-1\right)a=bx+2x-3b+7$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)a}{x-1}=\frac{bx+2x-3b+7}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$a=\frac{bx+2x-3b+7}{x-1}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax-2x-a-7}{3-x}\text{, }&x\neq 3\\b\in \mathrm{C}\text{, }&x=3\text{ and }a=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-3 by x.

$$ax-3x+3b-4=\left(b-1\right)x+a+3$$

Use the distributive property to multiply b-1 by x.

$$ax-3x+3b-4=bx-x+a+3$$

Subtract bx from both sides.

$$ax-3x+3b-4-bx=-x+a+3$$

Subtract ax from both sides.

$$-3x+3b-4-bx=-x+a+3-ax$$

Add 3x to both sides.

$$3b-4-bx=-x+a+3-ax+3x$$

Combine -x and 3x to get 2x.

$$3b-4-bx=2x+a+3-ax$$

Add 4 to both sides.

$$3b-bx=2x+a+3-ax+4$$

Add 3 and 4 to get 7.

$$3b-bx=2x+a+7-ax$$

Combine all terms containing b.

$$\left(3-x\right)b=2x+a+7-ax$$

The equation is in standard form.

$$\left(3-x\right)b=7+a+2x-ax$$

Divide both sides by -x+3.

$$\frac{\left(3-x\right)b}{3-x}=\frac{7+a+2x-ax}{3-x}$$

Dividing by -x+3 undoes the multiplication by -x+3.

$$b=\frac{7+a+2x-ax}{3-x}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{-bx-2x+3b-7}{x-1}\text{, }&x\neq 1\\a\in \mathrm{R}\text{, }&b=\frac{9}{2}\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-3 by x.

$$ax-3x+3b-4=\left(b-1\right)x+a+3$$

Use the distributive property to multiply b-1 by x.

$$ax-3x+3b-4=bx-x+a+3$$

Subtract a from both sides.

$$ax-3x+3b-4-a=bx-x+3$$

Add 3x to both sides.

$$ax+3b-4-a=bx-x+3+3x$$

Combine -x and 3x to get 2x.

$$ax+3b-4-a=bx+2x+3$$

Subtract 3b from both sides.

$$ax-4-a=bx+2x+3-3b$$

Add 4 to both sides.

$$ax-a=bx+2x+3-3b+4$$

Add 3 and 4 to get 7.

$$ax-a=bx+2x+7-3b$$

Combine all terms containing a.

$$\left(x-1\right)a=bx+2x+7-3b$$

The equation is in standard form.

$$\left(x-1\right)a=bx+2x-3b+7$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)a}{x-1}=\frac{bx+2x-3b+7}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$a=\frac{bx+2x-3b+7}{x-1}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax-2x-a-7}{3-x}\text{, }&x\neq 3\\b\in \mathrm{R}\text{, }&x=3\text{ and }a=\frac{13}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-3 by x.

$$ax-3x+3b-4=\left(b-1\right)x+a+3$$

Use the distributive property to multiply b-1 by x.

$$ax-3x+3b-4=bx-x+a+3$$

Subtract bx from both sides.

$$ax-3x+3b-4-bx=-x+a+3$$

Subtract ax from both sides.

$$-3x+3b-4-bx=-x+a+3-ax$$

Add 3x to both sides.

$$3b-4-bx=-x+a+3-ax+3x$$

Combine -x and 3x to get 2x.

$$3b-4-bx=2x+a+3-ax$$

Add 4 to both sides.

$$3b-bx=2x+a+3-ax+4$$

Add 3 and 4 to get 7.

$$3b-bx=2x+a+7-ax$$

Combine all terms containing b.

$$\left(3-x\right)b=2x+a+7-ax$$

The equation is in standard form.

$$\left(3-x\right)b=7+a+2x-ax$$

Divide both sides by -x+3.

$$\frac{\left(3-x\right)b}{3-x}=\frac{7+a+2x-ax}{3-x}$$

Dividing by -x+3 undoes the multiplication by -x+3.

$$b=\frac{7+a+2x-ax}{3-x}$$